Method of determining formation anisotropy in deviated wells using separation of induction mode

ABSTRACT

Measurements are made with a multicomponent induction logging tool in earth formations in a borehole inclined to earth formations. A combination of principal component measurements is used to determine the horizontal resistivity of the earth formations. The determined horizontal resistivities are used in a model for inversion of other components of the data to obtain the vertical formations resistivities. When multifrequency measurements are available, frequency focusing is used.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 09/825,104 filed on Apr. 3, 2001. It also claims priority fromU.S. Provisional Patent Application Ser. No. 60/312,655 filed on Aug.15, 2001. The contents of both documents are fully incorporated hereinby reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention is related generally to the field of interpretation ofmeasurements made by well logging instruments for the purpose ofdetermining the properties of earth formations. More specifically, theinvention is related to a method for determination of anisotropicformation resistivity in a deviated wellbore using multifrequency,multicomponent resistivity data.

2. Background of the Art

Electromagnetic induction and wave propagation logging tools arecommonly used for determination of electrical properties of formationssurrounding a borehole. These logging tools give measurements ofapparent resistivity (or conductivity) of the formation that whenproperly interpreted are diagnostic of the petrophysical properties ofthe formation and the fluids therein.

The physical principles of electromagnetic induction resistivity welllogging are described, for example, in, H. G. Doll, Introduction toInduction Logging and Application to Logging of Wells Drilled with OilBased Mud, Journal of Petroleum Technology, vol. 1, p. 148, Society ofPetroleum Engineers, Richardson Tex. (1949). Many improvements andmodifications to electromagnetic induction resistivity instruments havebeen devised since publication of the Doll reference, supra. Examples ofsuch modifications and improvements can be found, for example, in U.S.Pat. No. 4,837,517; U.S. Pat. No. 5,157,605 issued to Chandler et al,and U.S. Pat. No. 5,452,761 issued to Beard et al.

A limitation to the electromagnetic induction resistivity well logginginstruments known in the art is that they typically include transmittercoils and receiver coils wound so that the magnetic moments of thesecoils are substantially parallel only to the axis of the instrument.Eddy currents are induced in the earth formations from the magneticfield generated by the transmitter coil, and in the inductioninstruments known in the art these eddy currents tend to flow in groundloops which are substantially perpendicular to the axis of theinstrument. Voltages are then induced in the receiver coils related tothe magnitude of the eddy currents. Certain earth formations, however,consist of thin layers of electrically conductive materials interleavedwith thin layers of substantially non-conductive material. The responseof the typical electromagnetic induction resistivity well logginginstrument will be largely dependent on the conductivity of theconductive layers when the layers are substantially parallel to the flowpath of the eddy currents. The substantially non-conductive layers willcontribute only a small amount to the overall response of the instrumentand therefore their presence will typically be masked by the presence ofthe conductive layers. The non-conductive layers, however, are the onesthat are typically hydrocarbon bearing and are of the most interest tothe instrument user. Interpreting a well log made using theelectromagnetic induction resistivity well logging instruments known inthe art therefore may overlook some earth formations that might be ofcommercial interest.

The effect of formation anisotropy on resistivity logging measurementshas long been recognized. Kunz and Moran studied the anisotropic effecton the response of a conventional logging device in a boreholeperpendicular to the bedding plane of t thick anisotropic bed. Moran andGianzero extended this work to accommodate an arbitrary orientation ofthe borehole to the bedding planes.

Rosthal (U.S. Pat. No. 5,329,448) discloses a method for determining thehorizontal and vertical conductivities from a propagation or inductionwell logging device. The method assumes the angle between the boreholeaxis and the normal to the bedding plane, is known. Conductivityestimates are obtained by two methods. The first method measures theattenuation of the amplitude of the received signal between tworeceivers and derives a first estimate of conductivity from thisattenuation. The second method measures the phase difference between thereceived signals at two receivers and derives a second estimate ofconductivity from this phase shift. Two estimates are used to give thestarting estimate of a conductivity model and based on this model, anattenuation and a phase shift for the two receivers are calculated. Aniterative scheme is then used to update the initial conductivity modeluntil a good match is obtained between the model output and the actualmeasured attenuation and phase shift.

Hagiwara (U.S. Pat. No. 5,656,930) shows that the log response of aninduction-type logging tool can be described by an equation of the form$\begin{matrix}{V \propto {\frac{i}{L^{3}}\left( {{{- 2}{^{\quad k\quad L}\left( {1 - {i\quad k\quad L}} \right)}} + {i\quad k\quad {l\left( {^{\quad k\quad \beta} - ^{\quad k\quad L}} \right)}}} \right)}} & (1)\end{matrix}$

where

β²=cos²θ+λ² sin²θ  (2)

and

k ²=ω²μ(∈_(h) +iσ _(h)/ω)  (3)

where L is the spacing between the transmitter and receiver, k is thewavenumber of the electromagnetic wave, μ is the magnetic permeabilityof the medium, θ is the deviation of the borehole angle from the normalto the formation, λ is the anisotropy factor for the formation, ω is theangular frequency of the electromagnetic wave, σ_(h) is the horizontalconductivity of the medium and ∈_(h) is the horizontal dielectricconstant of the medium.

Eq. (3) is actually a pair of equations, one corresponding to the realpart and one corresponding to the imaginary part of the measured signal,and has two unknowns. By making two measurements of the measured signal,the parameters k and β can be determined. The two needed measurementscan be obtained from (1) R and X signals from induction logs, (2) phaseand attenuation measurements from induction tools, (3) phase orattenuation measurements from induction tools with two differentspacings, or (4) resistivity measurements at two different frequencies.In the low frequency limit, ∈ can be neglected in Eq. (3) and from knownvalues of ω and μ, the conductivity σ can be determined from k, assuminga value of μ equal to the permittivity of free space.

Wu (U.S. Pat. No. 6,092,024) recognized that the solution to eqs.(1)-(3) may be non-unique and showed how this ambiguity in the solutionmay be resolved using a plurality of measurements obtained from multiplespacings and/or multiple frequencies.

One solution to the limitation of the induction instruments known in theart is to include a transverse transmitter coil and a transversereceiver coil on the induction instrument, whereby the magnetic momentsof these transverse coils is substantially perpendicular to the axis ofthe instrument. Such as solution was suggested in Tabarovsky and Epov,“Geometric and Frequency Focusing in Exploration of Anisotropic Seams”,Nauka, USSR Academy of Science, Siberian Division, Novosibirsk, pp.67-129 (1972). Tabarovsky and Epov suggest various arrangements oftransverse transmitter coils and transverse receiver coils, and presentsimulations of the responses of these transverse coil systems configuredas shown therein. Tabarovsky and Epov also describe a method ofsubstantially reducing the effect on the voltage induced in transversereceiver coils which would be caused by eddy currents flowing in thewellbore and invaded zone. The wellbore is typically filled with aconductive fluid known as drilling mud. Eddy currents that flow in thedrilling mud can substantially affect the magnitude of voltages inducedin the transverse receiver coils. The wellbore signal reduction methoddescribed by Tabarovsky and Epov can be described as “frequencyfocusing”, whereby induction voltage measurements are made at more thanone frequency, and the signals induced in the transverse receiver coilsare combined in a manner so that the effects of eddy currents flowingwithin certain geometries, such as the wellbore and invasion zone, canbe substantially eliminated from the final result. Tabarovsky and Epov,however, do not suggest any configuration of signal processing circuitrywhich could perform the frequency focusing method suggested in theirpaper.

Strack et al. (U.S. Pat. No. 6,147,496) describe a multicomponentlogging tool comprising a pair of 3-component transmitters and a pair of3-component receivers. Using measurements made at two differentfrequencies, a combined signal is generated having a reduced dependencyon the electrical conductivity in the wellbore region. U.S. Pat. No.5,781,436 to Forgang et al, the contents of which are fully incorporatedherein by reference, discloses a suitable hardware configuration formultifrequency, multicomponent induction logging.

U.S. Pat. No. 5,999,883 issued to Gupta et al, (the “Gupta patent”), thecontents of which are fully incorporated here by reference, discloses amethod for determination of an initial estimate of the horizontal andvertical conductivity of anisotropic earth formations. Electromagneticinduction signals induced by induction transmitters oriented along threemutually orthogonal axes are measured at a single frequency. One of themutually orthogonal axes is substantially parallel to a logginginstrument axis. The electromagnetic induction signals are measuredusing first receivers each having a magnetic moment parallel to one ofthe orthogonal axes and using second receivers each having a magneticmoment perpendicular to a one of the orthogonal axes which is alsoperpendicular to the instrument axis. A relative angle of rotation ofthe perpendicular one of the orthogonal axes is calculated from thereceiver signals measured perpendicular to the instrument axis. Anintermediate measurement tensor is calculated by rotating magnitudes ofthe receiver signals through a negative of the angle of rotation. Arelative angle of inclination of one of the orthogonal axes that isparallel to the axis of the instrument is calculated, from the rotatedmagnitudes, with respect to a direction of the vertical conductivity.The rotated magnitudes are rotated through a negative of the angle ofinclination. Horizontal conductivity is calculated from the magnitudesof the receiver signals after the second step of rotation. An anisotropyparameter is calculated from the receiver signal magnitudes after thesecond step of rotation. Vertical conductivity is calculated from thehorizontal conductivity and the anisotropy parameter. One drawback inthe teachings of Gupta et al is that the step of determination of therelative angle of inclination of the required measurements of threecomponents of data with substantially identical transmitter-receiverspacings. Because of limitations on the physical size of the tools, thiscondition is difficult to obtain; consequently the estimates ofresistivities are susceptible to error. In addition, due to the highlynonlinear character of the response of multicomponent tools, suchinversion methods are time consuming at a single frequency and even moreso at multiple frequencies.

Analysis of the prior art leads to the conclusion that known methods ofdetermining anisotropic resistivities in real time require very lowfrequencies; as a consequence of the low frequencies, thesignal-to-noise ratio in prior art methods is quite low.

Co-pending U.S. patent application Ser. No. 09/825,104, (referred tohereafter as the '104 application) filed on Apr. 3, 2001 teaches acomputationally fast method of determination of horizontal and verticalconductivities of subsurface formations using a combination of dataacquired with a transverse induction logging tool such as the 3DEX^(SM)tool and data acquired with a conventional high definition inductionlogging tool (HDIL). The data are acquired in a vertical borehole. TheHDIL data are used to determine horizontal resistivities that are usedin an isotropic model to obtain expected values of the transversecomponents of the 3DEX^(SM). Differences between the model output andthe acquired 3DEX^(SM) data are indicative of anisotropy and thisdifference is used to derive an anisotropy factor. The method describedtherein has difficulties in deviated boreholes as the HDIL measurementsused to derive the isotropic model are responsive to both horizontal andvertical resistivity.

There is a need for a fast and robust method of determination ofanisotropic resistivity. Such a method should preferably be able to usehigh frequency measurements that are known to have bettersignal-to-noise ratio than low frequency methods. The present inventionsatisfies this need.

SUMMARY OF THE INVENTION

A method of logging subsurface formations using data acquired with atransverse induction logging tool, the formation having a horizontalconductivity and a vertical conductivity, by obtaining a plurality offrequencies measurements indicative of vertical and horizontalconductivities in a tool referenced coordinate system. The data aretransformed to a subsurface formation coordinate system. Multifrequencyfocusing is applied to the measurements at a plurality of frequencies.Horizontal formation conductivities are determined from a subset of thefocused conductivity measurements. Vertical formation conductivities aredetermined from the focused conductivity measurements associated withthe subsurface formation and the horizontal conductivities.

In a preferred embodiment of the invention, a transformation independentof the formation azimuth may be used to determine the conductivity ofthe transversely anisotropic formation.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1(Prior art) shows an induction instrument disposed in a wellborepenetrating earth formations.

FIG. 2 shows the arrangement of transmitter and receiver coils in apreferred embodiment of the present invention marketed under the name3DEX^(SM).

FIG. 3 shows an earth model example used in the present invention.

FIG. 4 is a flow chart illustrating steps comprising the presentinvention.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 (prior art) shows an induction well logging instrument 10disposed in a wellbore 2 penetrating earth formations. The earthformations are shown generally at 6, 8, 12 and 14. The instrument 10 istypically lowered into the wellbore 2 at one end of an armoredelectrical cable 22, by means of a winch 28 or similar device known inthe art. An induction well logging instrument which will generateappropriate types of signals for performing the process of the presentinvention is described, for example, in U.S. Pat. No. 5,452,761 issuedto Beard et al. The prior art induction logging tool includes atransmitter coil and a plurality of receiver coils 18A-18F. The coils inthe prior art device are oriented with the coil axes parallel to theaxis of the tool and to the wellbore.

Turning now to FIG. 2, the configuration of transmitter and receivercoils in a preferred embodiment of the 3DExplorer™ induction logginginstrument of Baker Hughes is disclosed. Such a logging instrument is anexample of a transverse induction logging tool. Three orthogonaltransmitters 101, 103 and 105 that are referred to as the T_(x), T_(z),and T_(y) transmitters are shown (the z-axis is the longitudinal axis ofthe tool). Corresponding to the transmitters 101, 103 and 105 arereceivers 107, 109 and 111, referred to as the R_(x), R_(z), and R_(y)receivers, for measuring the corresponding components (H_(xx), H_(yy),H_(zz)) of induced signals. In addition, cross-components are alsomeasured. These are denoted by H_(xy), H_(xz) etc.

FIG. 3 is a schematic illustration of the model used in the presentinvention. The subsurface of the earth is characterized by a pluralityof layers 201 a, 201 b, . . . , 201 i. The layers have thicknessesdenoted by h₁, h₂, . . . h_(i). The horizontal and verticalresistivities in the layers are denoted by R_(h1), R_(h2), . . . R_(hi)and R_(v1), R_(v2), . . . R_(vi) respectively. Equivalently, the modelmay be defined in terms of conductivities (reciprocal of resistivity).The borehole is indicated by 202 and associated with each of the layersare invaded zones in the vicinity of the borehole wherein borehole fluidhas invaded the formation and altered is properties so that theelectrical properties are not the same as in the uninvaded portion ofthe formation. The invaded zones have lengths L_(x01), L_(x02), . . .L_(x0i) extending away from the borehole. The resistivities in theinvaded zones are altered to values R_(x01), R_(x02), . . . R_(x0i). Inthe embodiment of the invention discussed here, the invaded zones areassumed to be isotropic while an alternate embodiment of the inventionincludes invaded zones that are anisotropic, i.e., they have differenthorizontal and vertical resistivities. It should further be noted thatthe discussion of the invention herein may be made in terms ofresistivities or conductivities (the reciprocal of resistivity). Thez-component of the 3DEX^(SM) tool is oriented along the borehole axisand makes an angle θ(not shown) with the normal to the bedding plane.The x-component of the tool makes an angle φ with the “up” direction.

In the '104 application to Tabarovsky, et al. multifrequency,multicomponent induction data are obtained using, for example, the 3DEX™tool, and a multifrequency focusing is applied to these data. Asdisclosed in U.S. Pat. No. 5,703,773 to Tabarovsky et al., the contentsof which are fully incorporated herein by reference, the response atmultiple frequencies may be approximated by a Taylor series expansion ofthe form: $\begin{matrix}{\begin{bmatrix}{\sigma_{a}\left( \omega_{1} \right)} \\{\sigma_{a}\left( \omega_{2} \right)} \\\vdots \\{\sigma_{a}\left( \omega_{m - 1} \right)} \\{\sigma_{a}\left( \omega_{m} \right)}\end{bmatrix} = {\begin{bmatrix}1 & \omega_{1}^{1/2} & \omega_{1}^{3/2} & \ldots & \omega_{1}^{n/2} \\1 & \omega_{2}^{1/2} & \omega_{1}^{3/2} & \ldots & \omega_{2}^{n/1} \\\vdots & \vdots & \vdots & ⋰ & \vdots \\1 & \omega_{m - 1}^{1/2} & \omega_{m - 1}^{3/2} & \ldots & \omega_{m - 1}^{n/2} \\1 & \omega_{m}^{1/2} & \omega_{m}^{3/2} & \ldots & \omega_{m}^{n/2}\end{bmatrix}\quad\begin{bmatrix}s_{0} \\s_{1/2} \\\vdots \\s_{{({n - 1})}/2} \\s_{n/2}\end{bmatrix}}} & (4)\end{matrix}$

In a preferred embodiment of the invention of the '104 application, thenumber m of frequencies ω is ten. In eq.(4), n is the number of terms inthe Taylor series expansion. This can be any number less than or equalto m. The coefficient s_(3/2) of the ω^(3/2) term (ω being the square ofk, the wave number) is generated by the primary field and is relativelyunaffected by any inhomogeneities in the medium surround the logginginstrument, i.e., it is responsive primarily to the formation parametersand not to the borehole and invasion zone. In fact, the coefficients_(3/2) of the ω^(3/2) term is responsive to the formation parameters asthough there were no borehole in the formation. Specifically, these areapplied to the H_(xx) and H_(yy) components. Those versed in the artwould recognize that in a vertical borehole, the H_(xx) and H_(yy) wouldbe the same, with both being indicative of the vertical conductivity ofthe formation. In one embodiment of the invention, the sum of the H_(xx)and H_(yy) is used so as to improve the signal to noise ratio (SNR).This multifrequency focused measurement is equivalent to the zerofrequency value. As would be known to those versed in the art, the zerofrequency value may also be obtained by other methods, such as byfocusing using focusing electrodes in a suitable device.

Along with the 3DEX™, the method of the '104 application also uses datafrom a prior art High Definition Induction Logging (HDIL) tool havingtransmitter and receiver coils aligned along the axis of the tool. Thesedata are inverted using a method such as that taught by Tabarovsky etal, or by U.S. Pat. No. 5,884,227 to Rabinovich et al., the contents ofwhich are fully incorporated herein by reference, to give an isotropicmodel of the subsurface formation. Instead of, or in addition to theinversion methods, a focusing method may also be used to derive theinitial model. Such focusing methods would be known to those versed inthe art and are not discussed further here. As discussed above, the HDILtool is responsive primarily to the horizontal conductivity of the earthformations when run in a borehole that is substantially orthogonal tothe bedding planes. The inversion methods taught by Tabarovsky et al andby Rabinovich et al are computationally fast and may be implemented inreal time. These inversions give an isotropic model of the horizontalconductivities (or resistivities)

Using the isotropic model derived, a forward modeling is used in the'104 application to calculate a synthetic response of the 3DEX™ tool ata plurality of frequencies. A suitable forward modeling program for thepurpose is disclosed in Tabarovsky and Epov “Alternating ElectromagneticField in an Anisotropic Layered Medium” Geol. Geoph., No. 1, pp.101-109. (1977). Multifrequency focusing is then applied to thesesynthetic data. In a preferred embodiment of the invention of the '104application, the method taught by Tabarovsky is used for the purpose.

In the absence of anisotropy, the output from model output should beidentical to the multifrequency focused measurements. Denoting byσ_(iso) the multifrequency focused transverse component synthetic datafrom and by σ_(meas) the multifrequency focused field data from, theanisotropy factor λ is then calculated in the '104 application.

The H_(xx) for an anisotropic medium is given by $\begin{matrix}{{H_{x\quad x} = {- {\frac{M}{4L^{3}}\left\lbrack {{- \left( \frac{L}{\delta_{v}} \right)^{2}} + {\left( {\frac{1}{3} + \frac{1}{\lambda}} \right)\left( \frac{L}{\delta_{h}} \right)^{3}}} \right\rbrack}}}{w\quad h\quad e\quad r\quad e}{{\delta_{v} = \sqrt{\frac{2}{{\omega\mu\sigma}_{v}}}},{\delta_{h} = \sqrt{\frac{2}{{\omega\mu\sigma}_{h}}}},{\lambda = {\frac{\sigma_{h}}{\sigma_{v}}.}}}} & (5)\end{matrix}$

For a three-coil subarray, $\begin{matrix}{H_{x\quad x} = {{- \frac{1}{4\pi}}\left( {\frac{1}{3} + \frac{1}{\lambda}} \right)\left( \frac{{\omega\mu\sigma}_{h}}{2} \right)^{3/2}{\sum M_{i}}}} & (6)\end{matrix}$

Upon introducing the apparent conductivity for H_(xx) this gives$\sigma_{m\quad e\quad a\quad s}^{3/2} = {\frac{3}{4}\left( {\frac{1}{3} + \frac{1}{\lambda}} \right)\sigma_{h}^{3/2}}$${{or}\left( {\sigma_{m\quad e\quad a\quad s}^{3/2} - \sigma_{i\quad s\quad o}^{3/2}} \right)} = {{\sigma_{h}^{3/2}\left( {\frac{1}{4} + \frac{3}{4\lambda} - 1} \right)} = {\sigma_{h}^{3/2}\left( {\frac{3}{4\lambda} - \frac{3}{4}} \right)}}$

which gives the result $\begin{matrix}{\lambda = \frac{1}{1 - {\frac{4}{3}\left( \frac{\sigma_{i\quad s\quad o}^{3/2} - \sigma_{m\quad e\quad a\quad s}^{3/2}}{\sigma_{t}^{3/2}} \right)}}} & (7)\end{matrix}$

where σ_(t) is the conductivity obtained from the HDIL data, i.e., thehorizontal conductivity. The vertical conductivity is obtained bydividing σ_(t) by the anisotropy factor from eq. (5). An importantaspect of the '104 application is that in a vertical borehole, themeasurements made by a HDIL tool depend only on the horizontalconductivities and not on the vertical resistivities. The method of theinvention disclosed there is to obtain an isotropic model from the HDILdata, use the isotropic model to predict measurements made on othercomponents and to use a difference between the predicted and actualmeasurements to obtain the vertical conductivity.

In a similar manner, the method of the present invention can be viewedas finding a combination of 3DEX^(SM) measurements (called modes of theinduction measurements) that are responsive only to the horizontalconductivity, deriving a model of horizontal conductivity from thiscombination of measurements, predicting values of other components of3DEX^(SM) measurements and using a difference between these predictedmeasurements and the actual measurements to determine a verticalconductivity. In a particular embodiment of the present invention, thedesired combination includes only the principal component measurements,i.e., upon H_(xx), H_(yy), and H_(zz). The flow chart of the method ofthe present invention is shown in FIG. 4

The method of the present invention starts with an estimate of the dipand azimuth of the formation relative to the borehole axis 300. Theseangles are defined below. In addition, a sensor on the logging tool alsoprovides another angle measurement called the “toolface angle” that isalso used in the analysis of the data. Multifrequency focused data arederived from multifrequency measurements 301. The data are transformed303 as discussed below to give measurements that are indicative only ofhorizontal conductivity. These data are inverted 305 to give a model ofthe horizontal conductivity of the data. These estimates of horizontalconductivity are used in an isotropic model as estimates of the verticalconductivity 307. The measured data are then inverted using this initialestimate of vertical conductivities 309. A check is made of the goodnessof fit (difference between a model output based on the inverted modeland the actual measurements) 311. If the difference is below apredetermined threshold, then the model is accepted. If the differenceis excessive, a iterative procedure is carried out with an updated angleestimate 313 until the result is acceptable. Any suitable iterativeprocedure may be used such as that based on a gradient method or amethod of steepest descent. Such iterative methods would be known tothose versed in the art and are not discussed further.

At this point we develop the principal component structure for measuringformation anisotropy in bedding planes when the borehole is not normal(perpendicular) to the bedding plane. For simplifying the notation, andto avoid confusion, the x-, y- and z-components in the tool coordinatesare called hereafter the (1, 2, 3) coordinate system. The x-, y- andz-components in the earth coordinate system will be referred to as such.

In the tool coordinate system, the matrix of magnetic components, H_(T),may be represented in the following form: $\begin{matrix}{{\hat{H}}_{T} = \begin{pmatrix}h_{11} & h_{12} & h_{13} \\h_{21} & h_{22} & h_{23} \\h_{31} & h_{32} & h_{33}\end{pmatrix}} & (8)\end{matrix}$

For layered formations, the matrix, H_(T), is symmetric. We measurethree diagonal elements, h₁₁, h₂₂, and h₃₃. The non-diagonal elementsare not needed in the present invention.

In the earth coordinate system, {x, y, z}, associated with the planeformation boundaries (z-axis is perpendicular to the boundaries anddirected downwards) the magnetic matrix may be presented as follows:$\begin{matrix}{{\hat{H}}_{M} = \begin{pmatrix}h_{x\quad x} & h_{x\quad y} & h_{x\quad z} \\h_{x\quad y} & h_{y\quad y} & h_{y\quad z} \\h_{x\quad z} & h_{y\quad z} & h_{z\quad z}\end{pmatrix}} & (9)\end{matrix}$

The formation resistivity is described as a tensor, ρ. In the coordinatesystem associated with a formation, the resistivity tensor has onlydiagonal elements in the absence of azimuthal anisotropy:$\begin{matrix}{\hat{\rho} = \begin{pmatrix}\rho_{t} & 0 & 0 \\0 & \rho_{t} & 0 \\0 & 0 & \rho_{n}\end{pmatrix}} & (10)\end{matrix}$

 ρ_(t)=ρ_(xx)=ρ_(yy), ρ_(n)=ρ_(zz)

The “tool coordinate” system (1-, 2-, 3-) can be obtained from the“formation coordinate” system (x-, y-, z-) as a result of two sequentialrotations:

(1) Rotation about the axis “2” by the angle θ, such that the axis “3”in a new position (let us call it “3′”) becomes parallel to the axis zof the “tool” system;

(2) Rotation about the axis “3′” by the angle Φ, such that the new axis“1” (let us call it “1′”) becomes parallel to the axis x of the toolsystem.

In the present invention, an iterative procedure as shown in FIG. 4 isused for θ and φ. θ is the relative inclination of the borehole axis tothe normal to the bedding while φ is the azimuth. An initial estimatefor θ and φ may be determined from borehole surveys or from resistivityimaging devices and from knowledge of the toolface angle.

The first rotation is described using matrices θ and θ^(T):$\begin{matrix}{{\hat{\theta} = \begin{pmatrix}C_{\theta} & 0 & S_{\theta} \\0 & 1 & 0 \\{- S_{\theta}} & 0 & C_{\theta}\end{pmatrix}},{{\hat{\theta}}^{T} = \begin{pmatrix}C_{\theta} & 0 & {- S_{\theta}} \\0 & 1 & 0 \\S_{\theta} & 0 & C_{\theta}\end{pmatrix}}} & (11)\end{matrix}$

Here, C_(θ)=cos θ, S_(θ)=sin θ

The second rotation is described using matrices Φ and Φ^(T):$\begin{matrix}{{\hat{\phi} = \begin{pmatrix}C_{\phi} & {- S_{\phi}} & 0 \\S_{\phi} & C_{\phi} & 0 \\0 & 0 & 1\end{pmatrix}},{{\hat{\phi}}^{T} = \begin{pmatrix}C_{\phi} & S_{\phi} & 0 \\{- S_{\phi}} & C_{\phi} & 0 \\0 & 0 & 1\end{pmatrix}}} & (12)\end{matrix}$

Here, C_(Φ)=cos Φ, S_(Φ)=sin Φ

Matrices H_(M) (the formation coordinate system) and H_(T) (the toolcoordinate system) are related as follows:

Ĥ ^(T) ={circumflex over (R)} ^(T) Ĥ _(m) {circumflex over (R)}  (13)

{circumflex over (R)} ^(T)={circumflex over (Φ)}^(T){circumflex over(θ)}^(T) , {circumflex over (R)}={circumflex over (θ)}Φ  (14)

It is worth noting that the matrix H_(M) contains zero elements:

h _(xy) =h _(yx)=0  (15)

This is true for multifrequency focused measurements as described below.It is also important to note that the following three components of thematrix H_(M) depend only on the horizontal resistivity.

h _(xz)=ƒ_(xz)(ρ_(t)), h _(yz)=ƒ_(yz)(ρ_(t)), h_(zz)=ƒ_(zz)(ρ_(t))  (16)

Two remaining elements depend on both horizontal and verticalresistivities.

h _(xx)=ƒ_(xx)(ρ_(t),ρ_(n)), h _(yy)=ƒ_(yy)(ρ_(t),ρ_(n))  (17)

Taking into account Equations (11), (12), (14) and (15), we can re-writeEquation (13) as follows: $\begin{matrix}\begin{matrix}{\begin{pmatrix}h_{11} & h_{12} & h_{13} \\h_{21} & h_{22} & h_{23} \\h_{31} & h_{32} & h_{33}\end{pmatrix} = \quad {\begin{pmatrix}C_{\phi} & S_{\phi} & 0 \\{- S_{\phi}} & C_{\phi} & 0 \\0 & 0 & 1\end{pmatrix}\begin{pmatrix}C_{\theta} & 0 & {- S_{\theta}} \\0 & 1 & 0 \\S_{\theta} & 0 & C_{\theta}\end{pmatrix}}} \\{\quad {\begin{pmatrix}h_{xx} & 0 & h_{xz} \\0 & h_{yy} & h_{yz} \\h_{xz} & h_{yz} & h_{zz}\end{pmatrix}\begin{pmatrix}C_{\theta} & 0 & S_{\theta} \\0 & 1 & 0 \\{- S_{\theta}} & 0 & C_{\theta}\end{pmatrix}}} \\{\quad \begin{pmatrix}C_{\phi} & {- S_{\phi}} & 0 \\S_{\phi} & C_{\phi} & 0 \\0 & 0 & 1\end{pmatrix}}\end{matrix} & (18)\end{matrix}$

The following expanded calculations are performed in order to presentEquation (18) in a form more convenient for analysis. $\begin{matrix}{{\hat{A}}_{1} = \quad {{\begin{pmatrix}C_{\theta} & 0 & S_{\theta} \\0 & 1 & 0 \\{- S_{\theta}} & 0 & C_{\theta}\end{pmatrix}\begin{pmatrix}C_{\phi} & {- S_{\phi}} & 0 \\S_{\phi} & C_{\phi} & 0 \\0 & 0 & 1\end{pmatrix}} = \begin{pmatrix}{C_{\theta}C_{\phi}} & {{- C_{\theta}}S_{\phi}} & S_{\theta} \\S_{\phi} & C_{\phi} & 0 \\{{- S_{\theta}}C_{\phi}} & {S_{\theta}S_{\phi}} & C_{\theta}\end{pmatrix}}} \\{{\hat{A}}_{2} = \quad {\begin{pmatrix}h_{xx} & 0 & h_{xz} \\0 & h_{yy} & h_{yz} \\h_{xz} & h_{yz} & h_{zz}\end{pmatrix}\begin{pmatrix}{C_{\theta}C_{\phi}} & {{- C_{\theta}}S_{\phi}} & S_{\theta} \\S_{\phi} & C_{\phi} & 0 \\{{- S_{\theta}}C_{\phi}} & {S_{\theta}S_{\phi}} & C_{\theta}\end{pmatrix}}} \\{= \quad \begin{pmatrix}{{C_{\theta}C_{\phi}h_{xx}} - {S_{\theta}C_{\phi}h_{xz}}} & {{{- C_{\theta}}S_{\phi}h_{xx}} + {S_{\theta}S_{\phi}h_{xz}}} & {{S_{\theta}h_{xx}} + {C_{\theta}h_{xz}}} \\{{S_{\phi}h_{yy}} - {S_{\theta}C_{\phi}h_{yz}}} & {{C_{\phi}h_{yy}} + {S_{\theta}S_{\phi}h_{yz}}} & {C_{\theta}h_{yz}} \\{{C_{\theta}C_{\phi}h_{xz}} + {S_{\phi}h_{yz}} - {S_{\theta}C_{\phi}h_{zz}}} & {{{- C_{\theta}}S_{\phi}h_{xz}} + {C_{\phi}h_{yz}} + {S_{\theta}S_{\phi}h_{zz}}} & {{S_{\theta}h_{xz}} + {C_{\theta}h_{zz}}}\end{pmatrix}} \\{{\hat{A}}_{3} = \quad {\begin{pmatrix}C_{\theta} & 0 & {- S_{\theta}} \\0 & 1 & 0 \\S_{\theta} & 0 & C_{\theta}\end{pmatrix}\begin{pmatrix}{{C_{\theta}C_{\phi}h_{xx}} - {S_{\theta}C_{\phi}h_{xz}}} & {{{- C_{\theta}}S_{\phi}h_{xx}} + {S_{\theta}S_{\phi}h_{xz}}} & {{S_{\theta}h_{xx}} + {C_{\theta}h_{xz}}} \\{{S_{\phi}h_{yy}} - {S_{\theta}C_{\phi}h_{yz}}} & {{C_{\phi}h_{yy}} + {S_{\theta}S_{\phi}h_{yz}}} & {C_{\theta}h_{yz}} \\{{C_{\theta}C_{\phi}h_{xz}} + {S_{\phi}h_{yz}} - {S_{\theta}C_{\phi}h_{zz}}} & {{{- C_{\theta}}S_{\phi}h_{xz}} + {C_{\phi}h_{yz}} + {S_{\theta}S_{\phi}h_{zz}}} & {{S_{\theta}h_{xz}} + {C_{\theta}h_{zz}}}\end{pmatrix}}}\end{matrix}$

The components of Â₃ are given as

a ₁₁ ⁽³⁾ =C _(θ) ² C _(Φ) h _(xx) −C _(θ) S _(θ) C _(Φ) h _(xz) −C _(θ)S _(θ) C _(Φ) h _(xz) −S _(θ) S _(Φ) h _(yz) +S _(θ) ² C _(Φ) h _(zz)

 [a ₁₁ ⁽³⁾ =C _(θ) ² C _(Φ) h _(xx)−2C _(θ) S _(θ) C _(Φ) h _(xz) −S_(θ) S _(Φ) h _(yz) +S _(θ) ² C _(Φ) h _(zz)](*

a ₁₂ ⁽³⁾ =−C _(θ) ² S _(Φ) h _(xx) +C _(θ) S _(θ) S _(Φ) h _(xz) +C _(θ)S _(θ) S _(Φ) h _(xz) −S _(θ) C _(Φ) h _(yz) −S _(θ) ² S _(Φ) h _(zz)

[a ₁₂ ⁽³⁾ =−C _(θ) ² S _(Φ) h _(xx)+2C _(θ) S _(θ) S _(Φ) h _(xz) −S_(θ) C _(Φ) h _(yz) −S _(θ) ² S _(Φ) h _(zz)](*)

a ₁₃ ⁽³⁾ =C _(θ) S _(θ) h _(xx) +C _(θ) ² h _(xz) −S _(θ) ² h _(xz) −C_(θ) S _(θ) h _(zz)

[a ₁₃ ⁽³⁾ =C _(θ) S _(θ) h _(xx)+(C _(θ) ² −S _(θ) ²)h _(xz) −C _(θ) S_(θ) h _(zz)](*)

[a ₂₁ ⁽³⁾ =S _(Φ) h _(yy) −S _(θ) C _(Φ) h _(yz)](*)

[a ₂₂ ⁽³⁾ =C _(Φ) h _(yy) +S _(θ) S _(Φ) h _(yz)](*)

[a ₂₃ ⁽³⁾ =C _(θ) h _(yz)](*)

a ₃₁ ⁽³⁾ =C _(θ) S _(θ) C _(Φ) h _(xx) −S _(θ) ² C _(Φ) h _(xz) +C _(θ)² C _(Φ) h _(xz) +C _(θ) S _(θ) h _(yz) −C _(θ) S _(θ) C _(Φ) h _(zz)

[a ₃₁ ⁽³⁾ =C _(θ) S _(θ) C _(Φ) h _(xx)+(C _(θ) ² −S _(θ) ²)C _(Φ) h_(xz) +C _(θ) S _(θ) h _(yz) −C _(θ) S _(θ) C _(Φ) h _(zz)](*)

a ₃₂ ⁽³⁾ =−C _(θ) S _(θ) S _(Φ) h _(xx) +S _(θ) ² S _(Φ) h _(xz) −C _(θ)² S _(Φ) h _(xz) +C _(θ) C _(Φ) h _(yz) +C _(θ) S _(θ) S _(Φ) h _(zz)

[a ₃₂ ⁽³⁾ =−C _(θ) S _(θ) S _(Φ) h _(xx)−(C _(θ) ² −S _(θ) ²)S _(Φ) h_(xz) +C _(θ) C _(Φ) h _(yz) +C _(θ) S _(θ) S _(Φ) h _(zz)](*)

 a ₃₃ ⁽³⁾ =S _(θ) ² h _(xx) +C _(θ) S _(θ) h _(xz) +C _(θ) S _(θ) h_(xz) +C _(θ) ² h _(zz)

[a ₃₃ ⁽³⁾ =S _(θ) ² h _(xx)+2C _(θ) S _(θ) h _(xz) +C _(θ) ² h _(zz)](*)

Taking into account all the above calculations, we can representEquation (18) in the following form: $\begin{pmatrix}h_{11} & h_{12} & h_{13} \\h_{21} & h_{22} & h_{23} \\h_{31} & h_{32} & h_{33}\end{pmatrix} = {\begin{pmatrix}C_{\phi} & S_{\phi} & 0 \\{- S_{\phi}} & C_{\phi} & 0 \\0 & 0 & 1\end{pmatrix}\begin{pmatrix}a_{11}^{3} & a_{12}^{3} & a_{13}^{3} \\a_{21}^{3} & a_{22}^{3} & a_{23}^{3} \\a_{31}^{3} & a_{32}^{3} & a_{33}^{3}\end{pmatrix}}$

The method of the present invention involves defining a linearcombination of the measurements that are responsive substantially to thehorizontal conductivity and not responsive to the vertical conductivity.In a preferred embodiment of the invention, the linear combination isthat of measurements h₁₁, h₂₂, and h₃₃ (i.e., the principal componentsonly), although in alternate embodiments of the invention, a linearcombination of any of the measurements may be used. The example givenbelow is that of the preferred embodiment.

Let us consider expressions for the measured principal components, h₁₁,h₂₂, and h₃₃: $\begin{matrix}\left\{ \begin{matrix}{h_{11} = {{a_{11}^{(3)}C_{\phi}} + {a_{21}^{(3)}S_{\phi}}}} \\{h_{22} = {{{- a_{12}^{(3)}}S_{\phi}} + {a_{22}^{(3)}C_{\phi}}}} \\{h_{33} = a_{33}^{(3)}}\end{matrix} \right. & (19)\end{matrix}$

More detailed representation yields:

h ₁₁ =C _(θ) ² C _(Φ) ² h _(xx)−2C _(θ) S _(θ) C _(Φ) ² h _(xz) −S _(θ)C _(Φ) S _(Φ) h _(yz) +S _(θ) ² C _(Φ) h _(zz) +S _(Φ) ² h _(yy) −S _(θ)C _(Φ) S _(Φ) h _(yz)

[h ₁₁ =C _(θ) ² C _(Φ) ² h _(xx) +S _(Φ) ² h _(yy)−2C _(θ) S _(θ) C _(Φ)² h _(xz)−2S _(θ) C _(Φ) S _(Φ) h _(yz) +S _(θ) ² C _(Φ) ² h_(zz)]  (20)

h ₂₂ =C _(θ) ² S _(Φ) ² h _(xx)−2C _(θi S) _(θ) S _(Φ) ² h _(xz) +S _(θ)C _(Φ) S _(Φ) h _(yz) +S _(θ) ² S _(φ) ² h _(zz) +C _(Φ) ² h _(yy) +S_(θ) C _(Φ) S _(Φ) h _(yz)

[h ₂₂ =C _(θ) ² S _(Φ) ² h _(xx) +C _(Φ) ² h _(yy)−2C _(θ) S _(θ) S _(Φ)² h _(xz)+2S _(θ) C _(Φ) S _(Φ) h _(yz) +S _(θ) ² S _(Φ) ² h_(zz)]  (21)

[h ₃₃ =S _(θ) ² h _(xx)+2C _(θ) S _(θ) h _(xz) +C _(θ) ² h _(zz)]  (22)

Expressions for each component, h₁₁, h₂₂, and h₃₃, contain two types offunctions: some depending only on ρ_(t), and some others depending onboth, ρ_(t) and ρ_(n). Equations (13)-(15) may be represented in thefollowing form: $\begin{matrix}\left\{ \begin{matrix}{h_{11} = {{C_{\theta}^{2}C_{\phi}^{2}h_{xx}} + {S_{\phi}^{2}h_{yy}} + {f_{11}\left( \rho_{t} \right)}}} \\{h_{22} = {{C_{\theta}^{2}S_{\phi}^{2}h_{xx}} + {C_{\phi}^{2}h_{yy}} + {f_{22}\left( \rho_{t} \right)}}} \\{h_{33} = {{S_{\theta}^{2}h_{xx}} + {f_{33}\left( \rho_{t} \right)}}}\end{matrix} \right. & (23) \\{{Here},} & \quad \\\left\{ \begin{matrix}{{f_{11}\left( \rho_{t} \right)} = {{{- 2}C_{\theta}S_{\theta}C_{\phi}^{2}h_{xz}} - {2S_{\theta}C_{\phi}S_{\phi}h_{yz}} + {S_{\theta}^{2}C_{\phi}^{2}h_{zz}}}} \\{{f_{22}\left( \rho_{t} \right)} = {{{- 2}C_{\theta}S_{\theta}S_{\phi}^{2}h_{xz}} + {2S_{\theta}C_{\phi}S_{\phi}h_{yz}} + {S_{\theta}^{2}S_{\phi}^{2}h_{zz}}}} \\{{f_{33}\left( \rho_{t} \right)} = {{2C_{\theta}S_{\theta}h_{xz}} + {C_{\theta}^{2}h_{zz}}}}\end{matrix} \right. & (24)\end{matrix}$

A linear combination of Equations (23) is defined in the form:

h=αh ₁₁ +βh ₂₂ +h ₃₃  (25)

Detailed consideration of Equation (25) yields:

h=αC _(θ) ² C _(Φ) ² h _(xx) +αS _(Φ) ² h _(yy) +αƒ ₁₁(ρ_(t))+βC _(θ) ²S _(Φ) ² h _(xx) +βC _(Φ) ² h _(yy)+βƒ₂₂(ρ_(t))+S _(θ) ² h_(xx)+ƒ₃₃(ρ_(t))

h=(αC _(θ) ² C _(Φ) ² +βC _(θ) ² S _(Φ) ² +S _(θ) ²)h _(xx)+(αS _(Φ) ²+βC _(Φ) ²)h _(yy)+αƒ₁₁(ρ_(t))+βƒ₂₂(ρ_(t))+ƒ₃₃(ρ_(t))

The method of the present invention involves defining the coefficients,α and β, in such a way that the resulting linear combination, h, doesnot depend on the vertical resistivity. To achieve that, we need to nullthe following part of the expression for h:

h _(ƒ)=(αC _(θ) ² C _(Φ) ² +βC _(θ) ² S _(Φ) ² +S _(θ) ²)h _(xx)+(αS_(Φ) ² +βC _(Φ) ²)h _(yy)=0  (26)

Imposing the following conditions satisfies equation (26):$\begin{matrix}\left\{ \begin{matrix}{{{\alpha \quad C_{\theta}^{2}C_{\phi}^{2}} + {\beta \quad C_{\theta}^{2}S_{\phi}^{2}} + S_{\theta}^{2}} = 0} \\{{{\alpha \quad S_{\phi}^{2}} + {\beta \quad C_{\phi}^{2}}} = 0}\end{matrix} \right. & (27)\end{matrix}$

Let us calculate the coefficients, α and β. The second Equation in (27)yields: $\begin{matrix}{\beta = {{- \frac{\quad S_{\phi}^{2}}{C_{\phi}^{2}}}\alpha}} & (28)\end{matrix}$

After substitution of Equation (28) in the first Equation of (27), weobtain: $\begin{matrix}\begin{matrix}\begin{matrix}{{{\alpha \quad C_{\theta}^{2}C_{\phi}^{2}} - {\left( {\frac{S_{\phi}^{2}}{C_{\phi}^{2}}\alpha} \right)C_{\theta}^{2}S_{\phi}^{2}} + S_{\theta}^{2}} = \quad {0 = {{\alpha \quad {C_{\theta}^{2}\left( {C_{\phi}^{2} - \frac{S_{\phi}^{4}}{C_{\phi}^{2}}} \right)}} + S_{\theta}^{2}}}} \\{\left. \Rightarrow\quad {\alpha \quad C_{\theta}^{2}\quad \frac{C_{\phi}^{4} - S_{\phi}^{4}}{C_{\phi}^{2}}} \right. = {- S_{\theta}^{2}}} \\{\left. \Rightarrow\quad {\alpha \quad C_{\theta}^{2}\quad \frac{\left( {C_{\phi}^{2} + S_{\phi}^{2}} \right)\left( {C_{\phi}^{2} - S_{\phi}^{2}} \right)}{C_{\phi}^{2}}} \right. = {- S_{\theta}^{2}}} \\{\left. \Rightarrow\quad {\alpha \quad C_{\theta}^{2}\frac{C_{2\phi}}{C_{\phi}^{2}}} \right. = {- S_{\theta}^{2}}}\end{matrix} \\{\alpha = {{- \frac{\quad C_{\phi}^{2}}{C_{2\phi}}}\frac{S_{\theta}^{2}}{C_{\theta}^{2}}}}\end{matrix} & (29)\end{matrix}$

To obtain the coefficient, β, let us substitute Equation (29) inEquation (28): $\begin{matrix}{\beta = {{\frac{S_{\phi}^{2}}{C_{\phi}^{2}}\frac{\quad C_{\phi}^{2}}{C_{2\phi}}\frac{S_{\theta}^{2}}{C_{\theta}^{2}}} = {\frac{\quad S_{\phi}^{2}}{C_{2\phi}}\frac{S_{\theta}^{2}}{C_{\theta}^{2}}}}} & (30)\end{matrix}$

Finally, $\begin{matrix}\left\{ \begin{matrix}{\alpha = {{- \frac{C_{\phi}^{2}}{C_{2\phi}}}\frac{S_{\theta}^{2}}{C_{\theta}^{2}}}} \\{\beta = {\frac{S_{\phi}^{2}}{C_{2\phi}}\frac{S_{\theta}^{2}}{C_{\theta}^{2}}}}\end{matrix} \right. & (31)\end{matrix}$

It is convenient to normalize coefficients, α and β. Let us introduce anormalization factor, κ.

κ={square root over (1+α²+β²)}  (32)

Equation (25) may be presented in the form:

h _(ƒ) =α′h ₁₁ +β′h ₂₂ +γ′h ₃₃  (33)

Here,

h _(ƒ′) =h _(ƒ)/κ, α′=α/κ, β′=β/κ, γ′=γ/κ.  (34)

Calculations yield:$\kappa^{2} = {{1 + {\frac{C_{\phi}^{4}}{C_{2\phi}^{2}}\frac{S_{\theta}^{4}}{C_{\theta}^{4}}} + {\frac{S_{\phi}^{4}}{C_{2\phi}^{2}}\frac{S_{\theta}^{4}}{C_{\theta}^{4}}}} = {{1 + {\frac{C_{\phi}^{4} + S_{\phi}^{4}}{C_{2\phi}^{2}}\frac{S_{\theta}^{4}}{C_{\theta}^{4}}}} = \frac{{C_{2\phi}^{2}C_{\theta}^{4}} + {\left( {C_{\phi}^{4} + S_{\phi}^{4}} \right)S_{\theta}^{4}}}{C_{2\phi}^{2}C_{\theta}^{4}}}}$

$\begin{matrix}{\kappa = \frac{\sqrt{{C_{2\phi}^{2}C_{\theta}^{4}} + {\left( {C_{\phi}^{4} + S_{\phi}^{4}} \right)S_{\theta}^{4}}}}{C_{2\phi}C_{\theta}^{2}}} & (35)\end{matrix}$

Consequently,$\gamma^{\prime} = \frac{C_{2\phi}C_{\theta}^{2}}{\sqrt{{C_{2\phi}^{2}C_{\theta}^{4}} + {\left( {C_{\phi}^{4} + S_{\phi}^{4}} \right)S_{\theta}^{4}}}}$$\alpha^{\prime} = {{{- \frac{C_{\phi}^{2}}{C_{2\phi}}}{\frac{S_{\theta}^{2}}{C_{\theta}^{2}} \cdot \frac{C_{2\phi}C_{\theta}^{2}}{\sqrt{{C_{2\phi}^{2}C_{\theta}^{4}} + {\left( {C_{\phi}^{4} + S_{\phi}^{4}} \right)S_{\theta}^{4}}}}}} = {- \frac{C_{\phi}^{2}S_{\theta}^{2}}{\sqrt{{C_{2\phi}^{2}C_{\theta}^{4}} + {\left( {C_{\phi}^{4} + S_{\phi}^{4}} \right)S_{\theta}^{4}}}}}}$$\beta^{\prime} = {{\frac{S_{\phi}^{2}}{C_{2\phi}}{\frac{S_{\theta}^{2}}{C_{\theta}^{2}} \cdot \frac{C_{2\phi}C_{\theta}^{2}}{\sqrt{{C_{2\phi}^{2}C_{\theta}^{4}} + {\left( {C_{\phi}^{4} + S_{\phi}^{4}} \right)S_{\theta}^{4}}}}}} = \frac{S_{\phi}^{2}S_{\theta}^{2}}{\sqrt{{C_{2\phi}^{2}C_{\theta}^{4}} + {\left( {C_{\phi}^{4} + S_{\phi}^{4}} \right)S_{\theta}^{4}}}}}$

Finally, we obtain: $\begin{matrix}\left\{ \begin{matrix}{\quad {\alpha^{\prime} = {- \frac{C_{\phi}^{2}S_{\theta}^{2}}{\kappa^{\prime}}}}} \\{\quad {\beta^{\prime} = \frac{S_{\phi}^{2}S_{\theta}^{2}}{\kappa^{\prime}}}} \\{\quad {\gamma^{\prime} = \frac{C_{2\phi}C_{\theta}^{2}}{\kappa^{\prime}}}}\end{matrix} \right. & (36)\end{matrix}$

Here,

κ′={square root over (C_(2Φ) ²C_(θ) ⁴+(C_(Φ) ⁴+S_(Φ) ⁴)S_(θ) ⁴)}  (37)

The coefficient, κ, degenerates under the following conditions:

θ=0, Φ=π/4=>κ′=0  (38)

Using the derivation given above, for an estimated value of θ and φ, theconductivities may be derived. A difference between the model output andthe measured values may then be used in the iterative proceduredescribed with respect to FIG. 4.

The derivation above has been done for a single frequency data.Multifrequency Focused (MFF) data is a linear combination of singlefrequency measurements so that the derivation given above is equallyapplicable to MFF data. It can be proven that the three principle 3DEX™measurements, MFF (multi-frequency focusing) processed, may be expressedin the following form: $\begin{matrix}{\begin{pmatrix}{{MFF}\left( h_{11} \right)} \\{{MFF}\left( h_{22} \right)} \\{{MFF}\left( h_{33} \right)}\end{pmatrix} = {\begin{pmatrix}a_{1} & a_{2} & a_{3} & a_{4} \\b_{1} & b_{2} & b_{3} & b_{4} \\c_{1} & c_{2} & c_{3} & c_{4}\end{pmatrix}\begin{pmatrix}{{MFF}\left( h_{x\quad x} \right)} \\{{MFF}\left( h_{y\quad y} \right)} \\{{MFF}\left( h_{z\quad z} \right)} \\{{MFF}\left( h_{x\quad z} \right)}\end{pmatrix}}} & (39)\end{matrix}$

The matrix coefficients of Equation 39 depend on θ_(r), φ_(r), and threetrajectory measurements: deviation, azimuth and rotation.

The components of the vector in the right hand side of Equation 39represent all non-zero field components generated by three orthogonalinduction transmitters in the coordinate system associated with theformation. Only two of them depend on vertical resistivity: h_(xx) andh_(yy). This allows us to build a linear combination of measurements,h₁₁, h₂₂, and h₃₃ in such a way that the resulting transformationdepends only on h_(zz) and h_(xz), or, in other words, only onhorizontal resistivity. Let T be the transformation with coefficients α,β and γ:

T=αMFF(h ₁₁)+βMFF(h ₂₂)+γMFF(h ₃₃)  (40)

The coefficients α, β and γ must satisfy the following system ofequations:

a ₁ α+b ₁ β+c ₁γ=0

a ₂ α+b ₂ β+c ₂γ=0  (41)

α²+β²+γ²=1

From the above discussion it follows that a transformation may bedeveloped that is independent of the formation azimuth. The formationazimuth-independent transformation may be expressed as:

T ₀=(h ₁₁ +h ₂₂)sin² θ−h ₃₃(1+cos²θ)  (42)

where θ is the dip of the formation and T₀ is the linear transformationto separate modes. With this transformation and the above series ofequations we may determine the conductivity of the transverselyanisotropic formation.

It is to be noted, however, that when the earth formation is uniform(i.e., there are no formation boundaries within the region ofinvestigation of the tool), it is not possible to satisfy eq. (40). Itis necessary then to have a measurement of at least one cross-component.

The present invention has been discussed above with respect tomeasurements made by a transverse induction-logging tool conveyed on awireline. This is not intended to be a limitation and the method isequally applicable to measurements made using a comparable tool conveyedon a measurement-while-drilling (MWD) assembly or on coiled tubing.

While the foregoing disclosure is directed to the preferred embodimentsof the invention, various modifications will be apparent to thoseskilled in the art. It is intended that all variations within the scopeand spirit of the appended claims be embraced by the foregoingdisclosure.

What is claimed is:
 1. A method of lagging a subsurface formationcomprising a plurality a layers each having a horizontal conductivityand a vertical conductivity, the method comprising: (a) conveying anelectromagnetic logging tool into a borehole in the subsurfaceformation, said logging tool including a plurality of transmitters and aplurality of receivers, at least one of said transmitters and at leastone of said receivers inclined to an axis of the tool, said boreholehaving an axis inclined at a nonzero angle to a normal to said layers;(b) using said electromagnetic logging tool for obtaining a plurality ofmeasurements with a plurality of pairs of said transmitters andreceivers; (c) using a first subset of said plurality of measurementsfor determining a horizontal conductivity associated with each of saidlayers; and (d) using determined horizontal conductivities and a secondsubset of said plurality of measurements for determining a verticalconductivity associated with each of said layers.
 2. The method of claim1 wherein said plurality of transmitters comprise x-, y- andz-transmitters and the plurality of receivers comprise x-, y- andz-receivers.
 3. The method of claim 2 wherein said first subset ofmeasurements consist of principal component measurements.
 4. The methodof claim 2 wherein determining the horizontal conductivity and thevertical conductivity associated with each of the plurality of layersfurther comprises obtaining a tool rotation angle, formation azimuth,and an angle of inclination of said borehole to the normal to theplurality of layers.
 5. The method of claim 2 wherein said subsurfaceformation further comprises a uniform formation, and the plurality ofmeasurements further comprises at least one measurement selected from(i) a h_(xz) measurement, (ii) a h_(xy) measurement, (iii) a h_(zx)measurement, (iv) a h_(zy) measurement, (v) a h_(yx) measurement, and,(vi) a h_(yz) measurement.
 6. The method of claim 1 wherein determiningthe horizontal conductivity associated with each of said layers furthercomprises applying frequency focusing to said first subset ofmeasurements and obtaining therefrom a first frequency focused set ofmeasurements.
 7. The method of claim 2 wherein determining thehorizontal conductivity associated with each of said layers furthercomprises applying frequency focusing to said first subset ofmeasurements and obtaining therefrom a second frequency focused set ofmeasurements.
 8. The method of claim 7 wherein determining thehorizontal conductivity associated with each of said layers furthercomprises determining a set of weights such that a weighted sum of thefirst frequency focused set of measurements is substantially independentof the vertical conductivity associated with each of the plurality oflayers.
 9. The method of claim 8 wherein determining the verticalconductivity associated with each of said layers further comprisesinverting the second frequency focused set of measurements using a modelincluding said horizontal and a vertical conductivity associated witheach of said plurality of layers.
 10. The method of claim 6 whereindetermining the horizontal conductivity associated with each of saidlayers further comprises determining a set of weights such that aweighted sum of the first frequency focused set of measurements issubstantially independent of the vertical conductivity associated witheach of the plurality of layers.
 11. The method of claim 10 whereindetermining the vertical conductivity associated with each of saidlayers further comprises inverting the second frequency focused set ofmeasurements using a model including said horizontal and a verticalconductivity associated with each of said plurality of layers.
 12. Themethod of claim 1 wherein determining the horizontal conductivity andthe vertical conductivity associated with each of the plurality oflayers further comprises obtaining a tool rotation angle, formationazimuth, and an angle of inclination of said borehole to the normal tothe plurality of layers.
 13. The method of claim 1 further comprisingrepeating (c)-(d) and iteratively updating an estimate of said non zeroangle until a difference between said measurements and a model outputobtained using said horizontal and vertical conductivities is less thana predetermined threshold.
 14. The method of claim 1 wherein determiningsaid horizontal conductivity associated with each of said layers furthercomprises performing an inversion.